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Topic: Working of a Polyphase Filter (Read 6829 times) previous topic - next topic
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Working of a Polyphase Filter

How does it work and it is the best way to devide a signal into subbands?
I´ve read the formula in this document (german, on page 11): http://www.ict.tuwien.ac.at/skripten/krypt...to/Mpgaudio.pdf
and took a look at the source of the polyphase filter code from musicdsp: http://www.musicdsp.org/archive.php?classid=3#39
Unfortunately the formual is not explained very good in the document, not even all variables are defined and the source is not commented 
Can someone here explain me the working of a polyphase filter (or a better filter deviding a signal into subbands)?

Working of a Polyphase Filter

Reply #2 –
Quote
Apart from the more traditional algorithms, wavelets can also be used to divide a signal into subbands. Here are two threads with some info on them:

OK, english is not my native language, it´s very difficult to read all these technically papers. In Valens´ wavelet guide there is a section about subband coding, the thing is called a "uniform iterated wavelet filterbank". But it doesn´t contain a formula or a description how it works. Except this name I don´t know anything more.
Also, what is the advantage (difference) of a wavelet filterbank over the "traditional algorithms" (what are they, and how do they work?), better quality or faster (or both)? Also, is that wavelet filterbank compatible to the "traditional algorithms", can the subband signals converted back to the main signal using a "traditional algorithm" with the same quality? Or is it better if there is "traditional algorithm" to decode, also use a traditional algorithm to encode the subbands?
Can you explain me the working of a iterated (wavelet) filterbank or point me somewhere, where this aplication is described?

Working of a Polyphase Filter

Reply #3 –
I don't really understand what the kind of explanation you want. (but if it's a good one I want it too  )

Wavelets are biortogonal, meaning that signal can be exactly reconstructed by the synthesis filter. Not all filters feature this. Wavelet filters are implemented in the following way: you have a lowpass and highpass filters, that together cover the full spectrum (the transition is at the middle, don't know if it's necessary though), and you use them recursevly, until you get the subbands you want. Pretty simple

-Eugene
The  greatest  programming  project of all took six days;  on the seventh  day  the  programmer  rested.  We've been trying to debug the !@#$%&* thing ever since. Moral: design before you implement.

Working of a Polyphase Filter

Reply #4 –
There are several types of wavelet transforms, but the one eltoder described is probably the easiest to understand. Usually the transform will be performed using floating point variables, but if you want to be able to perfectly reconstruct the signal you should use integers and a lifting wavelet transform.

Clemen’s Valens Homepage (‘A real friendly guide to wavelets’, ‘Fast Lifting Wavelet Transform’) has some information about the general concepts:
http://perso.wanadoo.fr/polyvalens/clemens...ns/clemens.html

Factoring Wavelet Transforms Into Lifting Steps has some information on the lifting transform as well and lists some coefficients.

The advantage of wavelets depends on what you want to do with them, if you simply want to put a frequency analysis on the screen it would be much easier to simply use one the standard FFT packages, but in some cases for example the ability to perfectly reproduce the signal can be helpfull, it's really a matter of what you want to do.

Working of a Polyphase Filter

Reply #5 –
Quote
I don't really understand what the kind of explanation you want. (but if it's a good one I want it too  )
The filter takes sample N, calculates XYZ, because .... and now he takes N+1... now we have output sample 1 in subband 0....
That would be a great explananation
Quote
Wavelets are biortogonal, meaning that signal can be exactly reconstructed by the synthesis filter. Not all filters feature this. Wavelet filters are implemented in the following way: you have a lowpass and highpass filters, that together cover the full spectrum (the transition is at the middle, don't know if it's necessary though), and you use them recursevly, until you get the subbands you want. Pretty simple
The theory is very simple. But how do I do that? Can a normal non-wavelet synthesis filter also reconstruct the signal?
Quote
The advantage of wavelets depends on what you want to do with them, if you simply want to put a frequency analysis on the screen it would be much easier to simply use one the standard FFT packages, but in some cases for example the ability to perfectly reproduce the signal can be helpfull, it's really a matter of what you want to do.
Devide a audio signal into uniform subbands, no FFT. Are wavelets here really the best thing to use, also when the synthesis filter is a traditional one?

Working of a Polyphase Filter

Reply #7 –
Quote
I don't really understand what the kind of explanation you want. (but if it's a good one I want it too  )

Wavelets are biortogonal, meaning that signal can be exactly reconstructed by the synthesis filter. Not all filters feature this. Wavelet filters are implemented in the following way: you have a lowpass and highpass filters, that together cover the full spectrum (the transition is at the middle, don't know if it's necessary though), and you use them recursevly, until you get the subbands you want. Pretty simple

-Eugene

Not all wavelets are biorthogonal and biorthgonality doesnt exactly mean it can be exactly reconstructed by the synthesis filter.

IIRC, orthogonal wavelet filters arent symmetrical and therefore do not possess the linear phase property (phase is important in image coding thus biorthogonal wavelets are almost always used).  Thus the orthogonality constraint was relaxed and a second wavelet was introduced which is orthogonal to the first.  ie. their inner product is the Kronecker delta. 

This leads to a departure from traditional subband coding where the QMFs consist of just one filter and you generate the high pass by shifting it by pi (inverting signs on alternate taps). 

But the wavelet transform using biorthogonal wavelet filters involve two filters.  The one's I've played with are the 9/7 tap spline invariant variety from Baulard, Antonini, et al.  where you swap the analysis filters at the synthesis end.

Just while on the topic, wavelet filters are one of the most amazing things ever invented from a subband point of view.  One of my colleagues tried to measure the SNR of a 'perfectly reconstructed' speech signal from an 8 tap QMF (I think it was Johnston's) and got only 48 dB.  I took the same signal and passed it through my 9/7 tap wavelet filter and got an SNR of 246 dB.  Now the total taps is the same (16) yet the wavelet filters did so much better!! 

(I must admit I cheated a teeny bit since I applied symmetric extension on the edges while my colleague was effectively extending with zeros.  But still, wavelets are better.  )

Working of a Polyphase Filter

Reply #8 –
Well... Orthogonality can be thought as a special case of biorthogonality I think.
Wim Sweldens says in this paper that lifting scheme fixes biorthogonality, and that's why it's so usefull, so I thought that this is the most important property.

Yep, I agree that wavelets are amazing, although I don't quite understand your comparasion. May be it's because I use integer wavelets

Also, have anyone tried to use overlapped windows with wavelets?

-Eugene
The  greatest  programming  project of all took six days;  on the seventh  day  the  programmer  rested.  We've been trying to debug the !@#$%&* thing ever since. Moral: design before you implement.

Working of a Polyphase Filter

Reply #9 –
Quote
This leads to a departure from traditional subband coding where the QMFs consist of just one filter and you generate the high pass by shifting it by pi (inverting signs on alternate taps). 
[a href="index.php?act=findpost&pid=148368"][{POST_SNAPBACK}][/a]

I just like the even order QMF filters...

Quote
Just while on the topic, wavelet filters are one of the most amazing things ever invented from a subband point of view.  One of my colleagues tried to measure the SNR of a 'perfectly reconstructed' speech signal from an 8 tap QMF (I think it was Johnston's) and got only 48 dB.  I took the same signal and passed it through my 9/7 tap wavelet filter and got an SNR of 246 dB.  Now the total taps is the same (16) yet the wavelet filters did so much better!!  
[a href="index.php?act=findpost&pid=148368"][{POST_SNAPBACK}][/a]

Well ... this is for sure no surprise since QMF filters do not satisfy the PR conditions perfectly. But you can design QMF filters with much lower reconstruction distortions (you need longer impulse responses).

/me wants biorthogonal linear phase even order wavelets - do they exist ?


Sebi